Consider the Laurent polynomial $f(t)=\sum \lambda_m t^m$. It is easy to prove that your conditions are equivalent to a system of $k+1$ equations $f^{(i)}(1)=0$ for $i=0,\dots,k$. In other words, $f(t)=(t-1)^{k+1}h(t)$ for some Laurent polynomial $h$ with integer coefficients. If $h(-1)\ne 0$, we get $|f(-1)|\geqslant 2^{k+1}$, this implies your claim. But if $h(-1)\ne 0$, it may fail, for example, consider $k=3$ and $f(t)=(t-1)(t^2-1)(t^3-1)$.

Consider the Laurent polynomial $f(t)=\sum \lambda_m t^m$. It is easy to prove that your conditions are equivalent to a system of $k+1$ equations $f^{(i)}(1)=0$ for $i=0,\dots,k$. In other words, $f(t)=(t-1)^{k+1}h(t)$ for some Laurent polynomial $h$ with integer coefficients. If $h(-1)\ne 0$, we get $|f(-1)|\geqslant 2^{k+1}$, this implies your claim. But if $h(-1)\ne 0$, it may fail, for example, consider $k=2$ and $f(t)=(t-1)(t^2-1)(t^3-1)$.